Nanoindentation relaxation study and micromechanics of Cement-Based Materials

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NANOINDENTATION RELAXATION STUDY AND

MICROMECHANICS OF CEMENT-BASED MATERIALS

Mémoire

Nicolas Venkovic

Maîtrise en génie civil

Maître ès sciences (M.Sc.)

Québec, Canada

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Résumé

Ce travail évalue le comportement mécanique des matériaux cimentaires à différentes échelles de distance. Premièrement, les propriétés mécaniques du béton produit avec un bioplastifi-ant à base de microorganismes efficaces (EM) sont etudiées par nanoindentation statistique, et comparées aux propriétés mécaniques du béton produit avec un superplastifiant ordinaire (SP). Il est trouvé que l’ajout de bioplastifiant à base de produit EM améliore la résistance des C–S–H en augmentant la cohésion et la friction des nanograins solides. L’analyse statistique des résultats d’indentation suggère que le bioplastifiant à base de produit EM inhibe la précip-itation des C–S–H avec une plus grande fraction volumique solide. Deuxièmement, un modèle multi-échelles à base micromécanique est dérivé pour le comportement poroélastique de la pâte de ciment au jeune age. L’approche proposée permet d’obtenir les propriétés poroélastiques requises pour la modélisation du comportoment mécanique partiellement saturé des pâtes de ciment viellissantes. Il est montré que ce modèle prédit le seuil de percolation et le module de Young non drainé de façon conforme aux données expérimentales. Un metamodèle stochas-tique est construit sur la base du chaos polynomial pour propager l’incertitude des paramètres du modèle à travers plusieurs échelles de distance. Une analyse de sensibilité est conduite par post-traitement du metamodèle pour des pâtes de ciment avec ratios d’eau sur ciment entre 0.35 et 0.70. Il est trouvé que l’incertitude sous-jacente des propriétés poroélastiques équiva-lentes est principalement due à l’énergie d’activation des aluminates de calcium au jeune age et, plus tard, au module élastique des silicates de calcium hydratés de basse densité.

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Abstract

This work assesses the mechanical behavior of cement-based materials through different length scales. First, the mechanical properties of concrete produced with effective microorganisms (EM)-based bioplasticizer are investigated by means of statistical nanoindentation, and com-pared to the nanomechanical properties of concrete produced with ordinary superplasticizer (SP). It is found that the addition of EM-based bioplasticizer improves the strength of C–S–H by enhancing the cohesion and friction of solid nanograins. The statistical analysis of indenta-tion results also suggests that EM-based bioplasticizer inhibits the precipitaindenta-tion of C–S–H of higher density. Second, a multiscale micromechanics-based model is derived for the poroelastic behavior of cement paste at early age. The proposed approach provides poroelastic properties required to model the behavior of partially saturated aging cement pastes. It is shown that the model predicts the percolation threshold and undrained elastic modulus in good agreement with experimental data. A stochastic metamodel is constructed using polynomial chaos ex-pansions to propagate the uncertainty of the model parameters through different length scales. A sensitivity analysis is conducted by post-treatment of the meta-model for water-to-cement ratios between 0.35 and 0.70. It is found that the underlying uncertainty of the effective poroelastic proporties is mostly due to the apparent activation energy of calcium aluminate at early age and, later on, to the elastic modulus of low density calcium-silicate-hydrate.

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6 Partial conclusion 59 II Uncertainty propagation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age 61 7 Partial introduction 65 8 Materials 67 9 From the general inclusion problem of Eshelby to microporomechanics 69 9.1 Generalized inclusion problem of Eshelby . . . 69

9.2 Homogenization scheme of Mori and Tanaka . . . 74

9.3 Self-consistent homogenization scheme . . . 80

9.4 Applications to microporomechanics . . . 81

10 A multiscale poromechanics-hydration model 87 10.1 Hydration model . . . 87

10.2 Multiscale poromechanics model . . . 90

11 Polynomial chaos expansion and post-processing 99 11.1 Polynomial chaos representation . . . 99

11.2 Post-processing . . . 101

12 Model input parameters 105 12.1 Phase composition . . . 105

12.2 Kinetic parameters . . . 106

12.3 Elastic parameters . . . 107

12.4 Microstructure parameters . . . 108

13 Results and discussion 109 13.1 Model validation . . . 109

13.2 Uncertainty propagation . . . 111

13.3 Global sensitivity analysis . . . 114

13.4 Probability density function . . . 116

14 Partial conclusion 121 Conclusion 123 A Matrix representation 125 Notation, conventions and identities 125 A.1 Matrix representation of tensors . . . 125

A.2 Matrix representation of tensorial operations . . . 126

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List of Tables

3.1 Mix designs of self-compacting concrete produced either with EM-based

bioplasti-cizer or superplastibioplasti-cizer . . . 12

4.1 Parameters of the shape functions for the Berkovich and spherical indenters. . . 16

5.1 Summary statistics of the cluster analysis based on the indentation modulus and hardness . . . 37

5.2 Converged strength properties assessed by inverse analysis of the packing density distributions . . . 45

5.3 Summary statistics of the cluster analysis of C–S–H phases for the sample with Mapefluid N-200 . . . 49

5.4 Summary statistics of the cluster analysis of C–S–H phases for the sample with IH Plus . . . 50

5.5 Macroscopic compressive strength . . . 57

8.1 Major oxides composition of cement PCCB9402 (Boumiz et al., 1996) . . . 67

12.1 Quantitative phase composition . . . 105

12.2 Kinetic parameters . . . 106

12.3 Apparent activation energies . . . 107

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List of Figures

1.1 Collapse of Koror-Babeldaob Bridge due to excessive delayed deformation, Republic of Palau (see Burgoyne and Scantlebury Burgoyne and Scantlebury (2006)). . . 2 4.1 Geometry of indentation tests with conical and spherical probes, see Vandamme

(2008). . . 15 4.2 Penetration depth history of the indenter. . . 18 4.3 Effect of viscosity on the measurement of contact stiffness. . . 21 4.4 Schematic representation of an indentation (adapted from Oliver and Pharr (1992)). 23 4.5 Energy activated process of a flow unit. . . 27 4.6 Vacancy migration of flow units as described by the Eyring model applied to viscous

flows. . . 27 4.7 Potential trough of a flow unit with and without shearing applied force. . . 28 4.8 Zhurkov dashpot and simple energy activated bodies. . . 31 5.1 Non-dimensional penetration and load relaxation curves measured on the sample

produced with Mapefluid N-200. . . 36 5.2 Non-dimensional penetration and load relaxation curves measured on the sample

produced with IH Plus. . . 37 5.3 Summary of the cluster analysis based on indentation modulus and hardness. (a)

Scatter-plot of indentation modulus and hardness with classification of indents and 90% confidence ellipsoids. (b) BIC values for different sizes of mixture. . . 38 5.4 Scatter-plots and classifications of (a) indentation modulus and (b) indentation

hardness with non-dimensional relaxation at 600 s. . . 40 5.5 Histogram of classification uncertainty for the indents attributed to cluster 4 or 5. 41 5.6 Scatter-plots and scaling of packing density with the indentation modulus. . . 43 5.7 Scatter-plots and scaling of packing density with the indentation hardness. . . 43 5.8 Scatter-plots and scaling of packing density with the activation volume. . . 44 5.9 Scatter-plots and scaling of packing density with the characteristic time of relaxation. 44 5.10 Convergence of the cohesion of the solid phase of C–S–H obtained by inverse

anal-ysis with an increasing number of indents. . . 46 5.11 Convergence of the friction coefficient of the solid phase of C–S–H obtained by

inverse analysis with an increasing number of indents. . . 47 5.12 Scatter-plots and scaling of the indentation modulus with the indentation hardness. 48 5.13 Scatter-plots and scaling of the indentation modulus with the activation volume. . 48 5.14 Scatter-plot and scaling of the indentation hardness with the activation volume. . . 49 5.15 Partial summary results of the cluster analysis based on indentation modulus,

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5.16 Partial summary results of the cluster analysis based on indentation modulus, hardness and activation volume. Model distributions of indentation modulus. . . . 51 5.17 Partial summary results of the cluster analysis based on indentation modulus,

hardness and activation volume. Model distributions of activation volume. . . 52 5.18 Partial summary results of the cluster analysis based on indentation modulus,

hardness and activation volume. BIC values for different sizes of mixtures. . . 52 5.19 Scatter-plots, classifications, confidence ellipsoids and scaling of the indentation

modulus with the packing density of the sample produced with Maplefluid N-200. . 53 5.20 Scatter-plots, classifications, confidence ellipsoids and scaling of the activation

vol-ume with the packing density of the sample produced with Maplefluid N-200. . . . 54 5.21 Scatter-plots, classifications, confidence ellipsoids and scaling of the indentation

modulus with the packing density of the sample produced with IH Plus. . . 54 5.22 Scatter-plots, classifications, confidence ellipsoids and scaling of the activation

vol-ume with the packing density of the sample produced with IH Plus. . . 55 5.23 Scatter-plots, classifications, confidence ellipsoids and scaling of the indentation

hardness with packing density. . . 56 9.1 2D heterogeneous representation of the Eshelby equivalent eigenstrain problem

with inelastic deformations. . . 70 9.2 2D homogeneous representation of the Eshelby equivalent eigenstrain problem with

inelastic deformations. . . 72 10.1 Multiscale representation of the microstructure of cement paste, adapted from

Con-stantinides (2002). . . 91 13.1 Model predictions of the volume fractions for w/c=0.50. . . 110 13.2 Model predictions of the undrained elastic modulus and experimental data from

Boumiz et al. (1996). . . 111 13.3 Model predictions of the undrained Poisson’s ratio and experimental data for

w/c=0.40 from Boumiz et al. Boumiz et al. (2000). . . 112 13.4 Prediction of the uncertainty of poroelastic properties of cement paste and pairwise

correlations. . . 113 13.5 Prediction of the uncertainty of poroelastic properties of cement paste and pairwise

correlations. . . 116 13.8 Pairwise correlations of the poroelastic properties with the percolation threshold. . 117 13.9 Sensitivity analysis of the percolation threshold - First Sobol’ indices. . . 117 13.10Sensitivity analysis of the drained elastic moduls - First order Sobol’ indices. . . . 118 13.11Sensitivity analysis of the Biot-Willis parameter - First order Sobol’ indices. . . 119 13.12Smoothed probability density function of the drained elastic modulus as a function

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Foreword

Two articles were inserted in this document. We give here the dates at which the articles were submitted, accepted and published. We also explain how the content of the originally published versions was adapted here. The role and contribution of the student is also described for each publication and some information is provided about the co-authors.

The first part of the document is a reproduction of an article that was submitted to Cement & Concrete Composites (Elsevier) the October 12th, 2012. The article was accepted after revision on December 6th, 2013 and eventually published as follows:

Venkovic N., Sorelli L. and Martirena F. (May 2014). Nanoindentation study of calcium silicate hydrates in concrete produced with effective microorganisms-based bioplasticizer. Cement & Concrete Composites, Volume 49: Pages 127–139.

The co-authors of the article are presented as follows. Luca Sorelli is associate professor at Laval Universty. Fernando-Martirena is director of the Center for Research and Development of Structures and Materials (CIDEM) at the University Central Marta Abreu of las Villas, Cuba. The student, first author of the article, proceeded to the surface preparation of the samples as well as the nanoindentation experiments on each sample. The student also analyzed the results and wrote the article. The samples were provided by Professor Martirena to Professor Sorelli. The methodology section of the article, here in Chapter 4, was extended to include additional information about the indentation analysis and the relaxation model calibrated experimentally. Also, several minor editions were made to the original text. The second part of the document is a reproduction of an article that was submitted to Prob-abilistic Engineering Mechanics (Elsevier) the January 3rd, 2012. The article was accepted after revision on December 18th, 2012 and eventually published as follows:

Venkovic N., Sorelli L., Sudret B., Yalamas T. and Gagné R. (April 2013). Uncertainty prop-agation of a multiscale poromechanics-hydration model for poroelastic properties of cement paste at early-age. Probabilistic Engineering Mechanics, Volume 32: Pages 5–20.

The other co-authors of the article are presented as follows. Bruno Sudret is professor at the Chair of Risk, Safety & Uncertainty Quantification of the Swiss Federal Institute of

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Tech-nology in Zurich (ETHZ), Switzerland. Thierry Yalamas is general manager at PHIMECA Engineering, France. Richard Gagné is professor at the University of Sherbrooke and adjunct director of the Research Center on Concrete Infrastructures (CRIB). The student, first author of the article, developed and implemented the poromechanical-hydration model. The student also performed the uncertainty and sensitivity analysis and wrote the article. The method-ology section of the article was extended through Chapter 9 where some elementary results of the problem of Eshelby are presented and applied to the derivation of common equations of microporomechanics. Also, some equations were corrected and several minor editions were made to the text.

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Chapter 1

Introduction

1.1 Problem Statement

Cement-based materials manufactured in deformable structures evolve over time and undergo delayed deformations. Even free of applied mechanical loads, concrete structures are subjected to thermal dilations and time-dependent volumetric deformations (shrinkage or swelling) due to internal chemical processes and changes in the water content of the material’s porous network. Beyond the extent of these processes, when a load is applied on a concrete structure, the underlying material deforms instantly and continues to deform over time (or creep) as the load is sustained. Conversely, sustained kinematic constraints lead up to stress releases (relaxation) in the material eventually triggering stress redistributions in a structure. We refer to these phenomena as time effects.

Time effects can affect the deflection of beams over long periods of time and increase the settlement of columns in tall buildings. When not properly taken into account, these phe-nomena can lead to structural disorders such as cracking, excessive deflection and differential settlement. Similarly, pre-stressed concrete structures such as beams, slabs, box girders and bridges sequentially built in stages can be widely affected by time-dependent deformations. Beyond their instantaneous mechanical response, these structures can undergo stress relax-ations inducing losses of pre-stress leading to the development of complex stress redistributions not considered in the original structural design. Pressure vessels and undersea shells are also highly sensitive to creep and shrinkage as they can trigger significant geometrical changes in the structure leading up to buckling and other instabilities. The Koror-Babeldaob (KB) Bridge used to connect two Pacific islands of the Republic of Paula is an infamous example of concrete structure that faced major disorders due to design mis-predictions of time effects. This bridge, built in 1977, was sequentially assembled by prestressing a box girder span of 241 m (Yee, 1979). Even from the begining of the construction, the structure was subjected to shrinkage, creep and loss of prestress leading to an increase of the midspan deflection over time. By 1990, the deflection reached 1.2 m and the serviceability of the bridge was

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compro-(a) Excessive deflection of KB Bridge (b) Collapse of KB Bridge

Figure 1.1: Collapse of Koror-Babeldaob Bridge due to excessive delayed deformation, Repub-lic of Palau (see Burgoyne and Scantlebury Burgoyne and Scantlebury (2006)).

mised (see Fig. 1.1a). In 1995, the deflection had increased so much that it was decided to perform remediation works in order to correct some of the sag and prevent further deflection (McDonald et al., 2004). In 1996, the bridge suddendly collapsed (see Fig. 1.1b) under neg-ligible traffic load and with no apparent external trigger (Pilz, 1997; McDonald et al., 2004; Burgoyne and Scantlebury, 2006). Even though the failure has still not been satisfactorily explained, it is clear that unexpected delayed deformations are at the origin of the structural disorders undergone by the bridge (Burgoyne and Scantlebury, 2006). The discrepancy be-tween the measurements and the delayed deflection computed for the design was pointed out by Bažant et al. (2008). Although these predictions were based on a model approved by the European Concrete Committee (European Concrete Committee (Comité européen du béton, CEB), 1972), the model of creep and shrinkage approved by the ACI Committee 209 (ACI Committee 209, 1972, 2008) would have predicted similar deformations (Bažant et al., 2008). Bažant et al. (2010) concluded that none of the current models for creep and shrinkage are satisfactorily predictive. The authors also highlighted the real necessity to improve creep (or relaxation) and shrinkage predictions from concrete composition.

1.2 Research Motivation

Predictive models for the behavior of concrete structures ideally rely on the numerical simulation of boundary value problems to satisfy field equations throughout some domain of interest. This approach can only yield satisfactory predictions if the constitutive mod-els assumed for every point in the domain are themselves reasonable abstractions of reality. Thereby, the path towards reliable predictions of structural time-effects starts with a predictive constitutive model.

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devel-opment of constitutive models. First, if one has enough information about the morphology and the mechanical behavior of the material, its mesostructure can be represented explicitly and a boundary value problem can be solved down to a resolution at which heterogeneities can not be discerned any more. Such an approach might be ideal if some limitations did not exist. First, cement-based materials are multiscale materials, meaning that the mechanical behavior of a piece of concrete subjected to a mechanical load is significantly affected by several behaviors occurring at different length scales. For this reason, the formulation of a constitutive model with finite computational resources is hardly a possibility. Moreover, cement-based materials are genuinely random systems of which the morphological and small scale constitutive details are only known to some degree. Thereby, lack of knowledge is another limiting factor of this approach. A second approach is to develop a set of mathematical equations between some state variables and loading conditions that satisfactorily reproduce the features of the relation between these quantities. Although the development of such empirical approaches can a priori form the basis of a predictive tool, it requires extensive experimental and/or numerical studies to reach an appropriate calibration. Also, once such a model is developed, it usually is only applicable to the very limited set of conditions under which the calibration was performed. Because cement-based materials are random media subjected to diverse types of solicitations, the resort to empirical methods offers only a small range of applications.

In the last 20 years or so, a different approach has started to emerge. This approach consists in formulating an abstraction of the mesostructure of a cement-based material parameterized in terms of meaningful morphological quantities likely to vary from a concrete mix to another. Some estimates of the state variables of the medium are then formulated through some averag-ing methods involvaverag-ing the morphological model assumed. This methodology relies on the idea that concrete, even when still hydrating, can be modeled as an ensemble of invariant material phases present in different proportions from one mix design to another and interacting with each other. This idea was mostly fostered by the work of Bažant (1977). An early use of this approach was made by Hua et al. (1997) to assess autogeneous shrinkage in aging cement-based materials. In the last 15 years, this methodology was developed considerably with the rise of more accessible nanoindentation facilities used to assess the invariant mechanical prop-erties of the elementary material phases of concrete, see Velez et al. (2001); Constantinides and Ulm (2004) and others. While almost systematically relying on micromechanics as an ap-plication of the results of Eshelby (1957), these methods have also been extended to account for the poroelastic component of the behavior of cement-based materials using results from microporomechanics (Dormieux et al., 2006a), see Ulm et al. (2004).

1.3 Research Objective

The resort to nanoindentation techniques along with applications of micromechanics has proven to be a synergistic method for at least two types of purposes. First, to assess whether

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or not changes of behaviors observed in concrete are triggered by (or correlated with) modi-fications of mechanical behaviors at small length scales of concrete. In this case, one usually assesses and compares the small scale mechanical properties of two types of samples known to exhibit distinct behaviors when subjected two mechanical loads. Instances of such analyses include the investigation of heat-treatment (Jennings et al., 2007; Vandamme et al., 2010), calcium leaching (Constantinides and Ulm, 2004) and thermal damage (DeJong and Ulm, 2007; Zanjani Zadeh and Bobko, 2013). A second purpose served by this methodology is the development of versatile models applied to materials with different compositions. An impor-tant example of such models is the one of Bernard et al. (2003) used to predict the early-age evolution over time of the elastic properties of an arbitrary mix design of concrete.

The objectives of the research presented in this document address each of the two purposes mentioned above. First, we want to illustrate the current state of the art methodology used for indentation analysis as a comparative tool for time effects in cement-based materials. To do so, we consider two distinct concrete materials; one produced with superplasticizer and the other with bioplasticizer. While the addition of bioplasticizer is known to enhance the resistance of concrete, it is our intent to (i) discover if this improvement is the result of a structural or me-chanical change at the nanoscale of concrete; and (ii) provide a first qualitative measure of the performance of materials produced with bioplasticizer in terms of relaxation when subjected to sustained loads. Notably, the resort to nanoindentation enables to survey the long-term relaxation behavior of calcium silicate hydrates (C–S–H) after few minutes only, while years-long experiments would be required to assess creep after conventional macroscopic methods. The second objective of this research is to understand how the uncertainty of the mechanical and morphological properties of the invariant material phases considered in micromechanical models affect the prediction of macroscopic state variables of interest to model time effects. If one understands clearly what random parameters of these models are responsible for the greatest source of uncertainty of model predictions, it can be legitimately decided to provide more efforts to obtain more accurate estimates of the corresponding properties.

1.4 Outline

The outline of the document is as follows. Part I is a comparative study conducted to as-sess whether changes of the behavior of concrete produced with bioplasticizer are triggered or not by mechanical and/or structural changes at the nanoscale. Part II presents a multiscale poromechanics-hydration model that could be used as a constitutive model for simulations intended to assess time effects in concrete structures. The uncertainty of the properties used as input parameters of the model are propagated up to the level of the predicted macro-scopic effective properties. A sensitivity is also conducted to identify which of these random parameters affect the most the predictions of the model.

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Part I

Nanoindentation study of calcium

silicate hydrates in concrete produced

with effective microorganisms-based

bioplasticizer

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Résumé

Les propriétés mécaniques du béton produit avec un bioplastifiant à base de microor-ganismes efficaces (EM) sont etudiées par nanoindentation statistique, et comparées aux propriétés mécaniques du béton produit avec un superplastifiant ordinaire (SP). Le recours à la nanoindentation permet une évaluation du comportement élastique, de la dureté et de la relaxation à long-terme des silicates de calcium hydratés (C–S–H) après seulement quelques minutes. Pour chaque matériau, une analyse de partitionnement de données révèle différents groupes d’indentations vraisemblablement effectuées sur des C–S–H avec fractions volumiques solides distinctes. Il est trouvé que l’ajout de bioplastifiant à base de produit EM améliore la résistance des C–S–H en augmentant la cohésion et la friction des nanograins solides, et réduit le taux absolu de relaxation à long-terme. L’analyse statistique des résultats d’indentation suggère que le bioplastifiant à base de produit EM inhibe la précipitation des C–S–H avec une plus grande fraction volumique solide. Ces observations corroborent les résultats d’une précédente étude qui attribuait une augmen-tation de l’homogénéité et un raffinement de la structure crystaline des phases de silicate à l’effet d’un agent similaire au bioplastifiant à base de produit EM. Il est aussi montré que l’amélioration des propriétés de résistance du C–S–H coincide avec un gain de résistance en compression mesuré à l’échelle macroscopique du béton à base de produit EM.

Abstract

The mechanical properties of concrete produced with effective microorganisms (EM)-based bioplasticizer are investigated by means of statistical nanoindentation, and com-pared to the nanomechanical properties of concrete produced with ordinary superplasti-cizer (SP). The resort to nanoindentation enables to survey the elasticity, hardness and long-term relaxation behavior of calcium silicate hydrates (C–S–H) after few minutes only. For each material, a cluster analysis of the experimental results yields groupings of indents likely performed on C–S–H with distinct packing densities. It is found that the addition of EM-based bioplasticizer improves the strength of C–S–H by enhancing the cohesion and friction of solid nanograins, and decreases the absolute rate of long-term relaxation. The statistical analysis of indentation results also suggests that EM-based bioplasticizer inhibits the precipitation of C–S–H of higher density. The findings of this work corrob-orate the results of a previous study which attributed an increase of homogeneity and a refinement of the crystalline structure of silicate phases to the effect of a biomodifier similar to EM-based bioplasticizer. The improvement of strength properties of C–S–H is also shown to coincide with a gain of compressive strength measured at the macroscale of EM-based concrete.

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Chapter 2

Partial introduction

The development of modern high performance concrete with outstanding mechanical proper-ties relies on an increased packing density of hydration products attainable by reducing the water-to-cement ratio (w/c) (Tennis and Jennings, 2000; Ulm et al., 2007). The drawback of decreasing the water-to-cement ratio of a concrete mix is a loss of workability generally compensated by the addition of expensive superplasticizer (SP) admixtures. Recently, the resort to low-cost bioplasticizers based on effective microorganisms (EM) (see Higa and Wi-didana (1991)) was found to be at least as efficient as SP for improving the workability of fresh concrete (Martirena et al., 2012). While EM have also proven to enable healing of concrete (Ramachandran et al., 2001; Wu et al., 2012) and strength improvement of mor-tar (Ghosh et al., 2005), no study has yet been performed which investigates the effects of EM-based admixtures on the nanomechanical properties of hardened concrete. More precisely, the properties of calcium silicate hydrates (C–S–H) are of capital interest. The relevance of studying C–S–H lies in the two following points: (i) their crystalline structure and variability throughout the system of hydrates may be affected by concrete admixtures such as EM-based bioplasticizers (Bolobova and Kondrashchenko, 2000); (ii) differences in their structure and properties can be responsible of important repercussions on macroscale phenomena such as drying shrinkage and long term creep (Jennings, 2000; Vandamme and Ulm, 2009). Therefore, the aim of this study is to provide an innovative insight of C–S–H mechanical properties in concrete produced with EM-based bioplasticizer. The investigation herein presented is focused on the elastic behavior, strength and long-term relaxation of C–S–H.

Two categories of concrete samples produced either with EM-based bioplasticizer or with or-dinary SP are investigated in order to compare the elastic response, hardness and long-term viscoelastic behavior of C–S–H. The characterization is made by nanoindentation for the fol-lowing reasons: (i) nanoindentation is a high resolution technique which enables to probe separately the distinct forms of C–S–H in concrete while macroscopic approaches are limited to the investigation of composite behaviors; (ii) the sharpness of an indenter tip is responsible for a high level of stress which shortens the duration of transient creep-relaxation by orders of

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magnitudes, hence allowing the characterization of a long-term viscoelastic behavior of C–S–H after few hundred seconds only (Vandamme and Ulm, 2009, 2013). Noteworthy, the long-term viscoelastic response of C–S–H to nanoindentation has shown to be quantitavely representative of the long-term creep observed after years of macroscale uniaxial creep (Vandamme and Ulm, 2013). Contrarily to previous investigations performed by nanoindentation (Vandamme, 2008; Vandamme and Ulm, 2009; Němeček, 2009; Jones and Grasley, 2011), the viscoelastic prop-erties are surveyed by relaxation (i.e. depth-controlled experiment) as it enables to record a material response which is not affected by time-delayed plastic deformation (Vandamme et al., 2012). Also in contrast to precedent studies (Constantinides, 2006; Ulm et al., 2007; Vanzo, 2009), the nanoindentation is performed on concrete samples. Thereby, the experimental re-sults are analyzed after a two-step cluster analysis with the objective to isolate the indents realized on C–S–H from the ones performed on phases which are out of the scope of this study. Following the current methodology of analysis of indentation results Ulm et al. (2007); Vandamme et al. (2010), the packing density distribution and nanoscale strength properties of C–S–H are identified in addition to the elastic stiffness and long-term rate of relaxation.

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Chapter 3

Materials

The EM-based bioplasticizer used in this study is produced at CIDEM-UCLV (Martirena et al., 2012). It consists of fallen leaves, rice, waste milk products, yogurt and molasses. Once the raw materials are mixed to form a solid substrate, it is left 25 days in a sealed tank for fermentation. The product is then used as inoculate to obtain a liquid fermentation of EM of which the pH is kept between 3.2 and 3.8. The resulting EM-based bioplasticizer is referred to as IH Plus. Meanwhile, the SP used in this study is a naphthalene-based product developed by MAPEI, Italy and referred to as Mapefluid N-200. Each plasticizer is used separately for a distinct sample. The sample preparation is processed after two steps. First, a bulk preparation consists in casting and curing the material and second, a surface preparation is done to precondition the sample for nanoindentation.

3.1 Bulk preparation

The samples used for this study consist of self-compacting concrete with mix designs given in Table 3.1. A type I cement (P-35 after Cuban norm NC 54 205:80) produced in Cienfuegos, Cuba is used with Cuban zeolite as a complementary source of fines (Martìnez-Ramìrez et al., 2006). The total amount of fines in the mix is 814 kg. A first sample is prepared with 40 L of IH Plus (6% of cement weight) and a second with 6.7 L of Mapelfuid N-200 (2% of cement weight). The amount of water is corrected in the mix with IH Plus in order to compensate for the excess of plasticizer. The water-to-fines ratio is kept around 0.56 for both mixes, and the fines to aggregates ratio is 0.72. The samples are cast in cubes of dimensions 100 x 100 x 100 mm and cured for one year. In the meantime, various properties of the fresh concrete are measured by Martirena et al. Martirena et al. (2012). At 365 days, some prisms of dimensions 4 x 4 x 1 cm are cut and kept for one week in isopropanol with two solvent replacements before being stored in desiccator for another week.

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Table 3.1: Mix designs of self-compacting concrete produced either with EM-based bioplasti-cizer or superplastibioplasti-cizer

Sample produced Cement Water Zeolite Sand Aggregate Plasticizer

with [kg] [L] [L] [L] [kg] [L]

Mapefluid N-200 450.0 459.0 364.0 594.6 528.0 6.7

IH Plus 450.0 426.0 364.0 594.6 528.0 40.0

3.2 Surface preparation

In order for nanoindentation results to relate to mechanical properties by classical contact analysis, the surface of the indented material should be flat (Oliver and Pharr, 1992). How-ever, due to a highly heterogeneous pore structure, cement-based materials exhibit significantly rough sections (Trtik et al., 2008). Thereby, in order to assess the results of surface prepa-ration, Miller et al. (2008) evaluated the surface roughness of finely polished cement paste at relevant scale for nanoindentation. For this purpose, a correction accounting for both the alignment of the sample and spatial waves of wavelength larger than 8 µm was applied before calculation of the surface roughness. A repeatable methodology was hence proposed which minimizes the surface roughness and ensures the convergence of the results of statistical de-convolution (Miller et al., 2008). Although no evidence was provided that the exposed pores are filled by detritus during surface preparation (Trtik et al., 2008), the issue of pore-filling remains an undiscussed question. Yet, similar protocols as the one of Miller et al. (2008) have been used for nanoindentation studies of cement-based materials (Chen et al., 2010; Davydov et al., 2011; Abuhaikal, 2011; Zanjani Zadeh and Bobko, 2013; Vallée et al., tion).

Before nanoindentation, the samples were cut into pieces of dimensions 1 x 1 x 0.8 cm. The surface of each sample was then prepared following the protocol of Vallée (2013) which yields surface roughnesses inferior to 20 nm for cement paste at different water-to-cement ratio. The surface preparation consists in two steps. First, the sample is glued with mounting wax on a steel disc and mounted on a jig. The jig is left 5 minutes on a polishing machine with an 8 inches silicon carbide disc (grain 240 grit). The position of the jig on the lapping wheel is such that the velocity of the disc underneath the sample remains between 18 cm/s to 25 cm/s. Second, the disc underneath the jig is replaced by a perforated polishing pad with 0.5 mL of 1 µm oil-based diamond suspension. The same lap frequency is kept for 4 hours. After polishing, the surface is gently cleaned with a soft cloth soaked with isopropanol.

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Chapter 4

Methods

A four-step methodology is adopted to assess and compare the elastic and nanoscale strength properties, packing density distribution and long-term relaxation of C–S–H in both samples produced either with EM-based bioplasticizer or with SP.

• First, a large number of depth-controlled nanoindentation experiments is performed on both materials under study. Although the indents are preferably realized on bulk cement paste; sand grains, interfacial transition zones (ITZ) and aggregates are also probed unintentionally. The purpose of this step is to collect enough local observations to provide a statistically accurate picture of the nanomechanical behavior of C–S–H. • Second, in an attempt to isolate the indents which where performed on C–S–H, all

the indentation results are merged into a common data set irrelevantly of the sample they were measured on. This enables to increase the number of indents performed on phases which behave similarly in both materials and were sparsely sampled because unintentionally indented. The phases of concern are aggregates, sand grains, unhydrated cement grains and, arguably, large crystals of portlandite. A cluster analysis of the merged data set is thus performed with respect to the indentation elastic modulus and hardness. As a result, the indents performed on the material phases which are out of the scope of this study are identified and removed from the data set. At this point, mostly results of indentation performed on C–S–H phases packed after distinct densities remain.

• Third, the remaining indentation results are split back in two data sets accordingly to the sample they were measured on. Then, the packing density distribution and strength properties of C–S–H are assessed for each sample following the same procedure as described by Ulm et al. (2007) and Vandamme et al. (2010). For every indent, the observed stress relaxation is characterized by a characteristic time of relaxation and an activation volume which are representative of the viscoelastic behavior described after

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an energy-activated (Klug and Wittmann, 1974; Wittmann, 1982) rearranging process of elementary particles of C–S–H (Ulm et al., 2000; Jennings, 2004; Vandamme and Ulm, 2009).

• Fourth, a cluster analysis of the indentation modulus, hardness and activation volume of relaxation is performed to identify the different phases among the remaining data of each sample. Different C–S–H phases and their distinct mechanical properties are identified for both samples produced either with IH Plus or with Mapefluid N-200.

The homogeneity of the indented microvolumes assumed in nanoindentation studies (Constan-tinides and Ulm, 2007; Ulm et al., 2007; Sorelli et al., 2008; Vandamme et al., 2010) has been questioned by Trtik et al. (2009) and Lura et al. (2011). While the extent of this probable limitation is still being debated (Ulm et al., 2010; Davydov et al., 2011), the resort to nanoin-dentation along with micromechanics was conclusive enough to lead up to a similar stiffness of C–S–H grains (Constantinides and Ulm, 2007) as what was later assessed by molecular dynamics (Pellenq et al., 2009). A promising approach to circumvent potential issues related to the size of the investigated microvolume is the use of peak-force tapping atomic force mi-croscopy (AFM), see Trtik et al. (2012). However, contrarily to statistical nanoindention (Ulm et al., 2007; Bobko et al., 2011; Vandamme and Ulm, 2013), this method neither yet enables to assess strength properties nor can it measure time-delayed mechanical behaviors. Also, al-though AFM can be used to quickly perform myriads of measurements, it was only used over small regions of cement paste so that the full range of hydrates was never investigated with this sole technique (Trtik et al., 2012). Meanwhile, nanoindentation has already demonstrated some versatility for the comparative assessment of mechanical properties of hydrates with the intent to determine whether changes of behaviors observed at macroscales were triggered by (or correlated with) modifications of mechanical behaviors at nanoscales. Instances of such analyses include the investigation of heat-treatment (Jennings et al., 2007; Vandamme et al., 2010), calcium leaching (Constantinides and Ulm, 2004) and thermal damage (DeJong and Ulm, 2007; Zanjani Zadeh and Bobko, 2013). Thereby, nanoindentation is currently the most suited technique to compare the stiffness, strength and long-term time-delayed mechanical behavior of hydrates in concrete produced either with EM-based bioplasticizer or with SP. The technical aspects invoked through the four-step methodology adopted in this study are summarized in the next sections.

4.1 Nanoindentation relaxation analysis

Indentation consists in pressing a reference punch, i.e. indenter, at the surface of a deformable medium, see Fig. 4.1. By means of a proper analysis of the outputs of an indentation, one can extract information about the mechanical behavior and properties of the indented material.

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Figure 4.1: Geometry of indentation tests with conical and spherical probes, see Vandamme (2008).

A most common indentation analysis requires some knowledge of the mechanical properties of the indenter as well as the histories of the applied load, penetration depth and contact area of the indented surface. Here, the indenter is considered infinitely rigid compared to the indented material.

4.1.1 Self-similarity

Indentation analysis is performed by applying the solution of a quasi-static contact problem between two solid bodies. The greatest source of difficulty in the analysis comes from the fact that the contact area between the indenter and the specimen is not known a priori. As a mean to simplify the resolution of the associated contact problem, one can perform a self-similar indentation; meaning that the displacement field at any load P2 can be inferred from the

known displacement field at a distinct load P1. Borodich et al. (2003) stated three conditions

that must be satisfied for an indentation to be self-similar:

1. The constitutive behavior of the indented material is homogeneous with respect to strains or stresses. This is satisfied if the operator F of the constitutive relation between strains and stresses is a homogeneous function of degree κ:

λκ_{F (ε) = F (λε)} _(4.1)

where ε is the infinitesimal strain tensor and λ is an arbitrary constant. The constitutive relationship of non-aging viscoelastic media such as mature cement-based materials satisfies this condition. Indeed, the stress state of such media is given by

λσ(t) = λ Z t 0 C(t− τ) : dε dτ dτ = Z t 0 C(t− τ) : d dτ(λε) dτ (4.2)

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Table 4.1: Parameters of the shape functions for the Berkovich and spherical indenters.

Probe type d B θeq R

Berkovich 1 cot θeq 70.32◦

-Spherical 2 (2R)−1 _- _{10 µm}

where C(t) is a time-dependent stiffness tensor so that the constitutive behavior is homoge-neous of degree one.

2. The shape of the indenter can be described by a homogeneous function of degree d greater than or equal to one. Since it is common either to perform indentation with an axisymmetric punch or, to use a probe with a shape that can reasonably be described by an axisymmet-ric function, it is useful to formulate this condition with respect to a cylindaxisymmet-rical system of coordinates (r, z, θ):

x· ez= B [x· er] rd for every x at the edge of the probe (4.3)

where B is referred to as a proportionality factor B, see Fig. 4.1. Two different probes whose geometries can be described or approximated after Eq. (4.3) are commonly used for indentation studies. Those are:

• The Berkovich indenter which consists in a sharp three sided pyramidal tip used to probe areas as small as few hundreds square nanometers, i.e. nanoindentation. The geometry of this probe induces a high concentration of stresses beneath the contact zone that results in important plastic deformations. To simplify the contact analysis, the pyramidal shape is represented by an equivalent cone of half-angle θeq (see Table 4.1).

• The spherical indenter of radius R used to reduce the amount of induced plastic defor-mations.

The proportionality factor B and the degree d of the homogeneous shape functions associated with these two indenters are given in Table 4.1.

3. The load (or penetration) at the surface of the specimen increases monotonically during in-dentation. The condition of monotonic increase is not satisfied at load (or penetration) release so that self-similarity can not be invoked to analyze the unloading phase of an indentation experiment.

4.1.2 Scaling relations

If self-similarity is satisfied, the load P2 applied at the surface penetrated by the indenter at

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surface at a depth h1. Therefore, considering that the state of a self-similar indentation is

fully characterized by the load P , a penetration depth h, a contact depth hc (see Fig. 4.1)

and a projected contact area Ac(see Fig. 4.1), the following scaling relations apply (Borodich,

1989): P1 P2 = h1 h2 2+κ(d−1) d (4.4) h1 h2 = (Ac)1 (Ac)2 d/2 (4.5) where κ is equal to one for linear viscoelastic media. For axisymmetric indenters, the contact area Ac is a disk of radius a so that Eq. (4.5) can be recast in

ad 1 h1 = a d 2 h2 . (4.6)

Considering that the profile of the indenter is described by Eq. (4.3), it can be shown from Eq. (4.6) that the ratio of contact over penetration depth does not depend on the applied load:

(hc)1

h1

= (hc)2 h2

. (4.7)

In other words, the ratio hc/hremains constant during indentation until the sustained

pene-tration is released.

4.1.3 Definition of the contact problem with relaxation

For this study, we consider depth-controlled experiments for which the intented material ex-hibits some stress relaxation. The prescribed penetration history is described by a function of the form

h(t) =_F(t)hmax (4.8)

where hmaxis a constant penetration depth sustained during relaxation and F(t) is referred to

as a history function with property max F = 1. The following trapezoidal penetration history (see Fig. 4.2) is considered:

F(t) =      t/τL for 0 ≤ t ≤ τL 1 for τL≤ t ≤ τL+ τH (τL+ τH+ τU− t)/τU for τL+ τH ≤ t ≤ τL+ τH + τU (4.9) where τL is the duration of the loading phase, τH is the time during which the maximal

penetration depth is sustained and τU is the unloading time of the specimen.

The indented material is considered as a solid half-space Ω(t) with boundary ∂Ω(t) = ∂Ωc(t)∪

∂Ωnc(t)in which ∂Ωc(t)is the surface of contact with the probe and ∂Ωnc(t)is the part of the

boundary which is not in contact with the indenter. We note n(x, t) and t(x, t) the outward normal and tangent vectors at a point x of the boundary ∂Ω(t).

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Figure 4.2: Penetration depth history of the indenter.

The assumption is made that the deformable medium undergoes infinitesimally small defor-mations only so that the strain tensor is given by

ε(t) = 1 2

∇u(x, t) +t∇u(x, t) ∀ x ∈ Ω(t), ∀ t (4.10) in which u is the displacement field at time t. Another assumption is made that every point within the indented solid remains under quasi-static equilibrium:

∇ · σ(x, t) = 0 ∀ x ∈ Ω(t), ∀ t. (4.11)

The boundary points of Ω(t) which are not in contact with the indenter remain traction-free during the whole experiment:

σ(x, t)_{· n(x, t) = 0} _{∀ x ∈ ∂Ω}nc(t),∀ t (4.12)

while the contact between the indenter and the deformable solid is assumed friction-free: t(x, t)_{· σ(x, t) · n(x, t) = 0} _{∀ x ∈ ∂Ω}c(t),∀ t. (4.13)

Considering the parametric representation of the contour of the rigid indenter, see Eq. (4.3), the only non-vanishing component of the displacement field is given as follows

u(x, t)_{· e}_z = h(t)_{− B [x · e}_r]d _{∀ x ∈ ∂Ω}c(t),∀ t (4.14)

onto the moving contact boundary. The global equilibrium of the solid half-space is guaranteed through the following equation:

P (t) = Z

∂Ωc(t)

e_z_{· σ(x, t) · n(x, t) dA} _{∀ t.} (4.15)

4.1.4 Elastic solution of Galin-Sneddon

Analytical approximations to solutions of self-similar indentation boundary value problems are often built upon the solutions of Galin (1953) and Sneddon (1965). Assuming an isotropic

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elastic deformable medium, the following relation was established between the load applied at the surface of a specimen and the penetration depth of the indenter:

P = d d + 1 Γ(d/2 + 1/2) Γ(d/2 + 1) 1/d 2h1+1/d (√πB)1/d M0 (4.16)

where Γ(x) is the Euler Gamma function1_{. The indentation elastic modulus M}

0 is given by M0= E0 (1_{− ν}0)2 = 4G0 3K0+ G0 3K0+ 4G0 (4.17) where E0 is the elastic modulus, ν0 is the Poisson’s ratio and, K0 and G0 are the elastic bulk

and shear moduli. It is also common practice to extract the elastic indentation modulus M0

from the measurement of a contact stiffness through the BASh formula (Bulychev et al., 1975): S0≡ dP dh t=(τL+τH)+ = √2 π p AcM0 (4.18)

where the elastic contact stiffness S0 is the derivative of the applied load with respect to the

penetration depth at the onset of unloading. Still following the solution of Galin (1953) and Sneddon (1965), the invariant ratio of contact over penetration depth is given by

Λ≡ hc h = 1 √ π Γ(d/2 + 1/2) Γ(d/2 + 1) = ( 2/π for d = 1 1/2 for d = 2 . (4.19)

4.1.5 Approximate solution for indentation with relaxation

Although the analytical relation of Galin-Sneddon is only valid for elastic solids, it is frequently used to analyze results of nanoindentation despite the occurrence of plastic deformations dur-ing penetration of the indenter. Thereby, neglectdur-ing first the contribution of plastic defor-mations, Eq. (4.16) is recast in the following convolution integral to account for relaxation (Radok, 1957; Lee and Radok, 1960):

P (t) = 2 d d + 1 Λ B 1/dZ t 0 M (t_{− τ)}d{h(τ)} 1+1/d dτ dτ. (4.20)

where the indentation modulus M(t) becomes a decreasing function of time. Using the previ-ous formulation of penetration history (see Eq. (4.8)), the applied load is recast in

P (t) = 2 h1+1/d_max d d + 1 Λ B 1/dZ t 0 M (t_{− τ)}d{F(τ)} 1+1/d dτ dτ. (4.21)

Considering the piecewise definition of the controlled penetration, the expression of the applied load is given as follows for the loading, the holding and the unloading phase:

P (t) =      PL(t) for 0 ≤ t ≤ τL PH(t) for τL≤ t ≤ τL+ τH PU(t) for τL+ τH ≤ t ≤ τL+ τH + τU (4.22) 1

The Euler Gamma function is given by Γ(x) =

∞

R

0

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where the applied load of the loading phase is PL(t) = 2 h1+1/dmax Λ B 1/dZ t 0 M (t_{− τ)} τL τ τL 1/d dτ, (4.23)

the applied load of the holding phase reads PH(t) = 2 h1+1/dmax Λ B 1/dZ τL 0 M (t− τ) τL τ τL 1/d dτ (4.24)

and, at the onset of unloading, the applied load is PU(t) = 2 h1+1/dmax Λ B 1/d  τL Z 0 M (t− τ) τL τ τL 1/d dτ− t Z τL+τH M (t_{− τ)} τU τT − τ τU 1/d dτ   (4.25)

where τT is the overall time length of indentation (τL+ τH + τU).

As stated by Eq. (4.18), the contact stiffness is the derivative of the applied load with respect to the penetration depth at the onset of unloading. For the given function of penetration history, this becomes

S = dP dh _t=(τ L+τH)+ = P (t)˙ ˙h(t) t=(τL+τH)+ = P˙U(t = (τL+ τH) +₎ ˙ F (t = (τL+ τH)+) hmax . (4.26)

For a more accurate evaluation of the elastic indentation modulus, one should take into account the contribution of viscous effects on the measurement of the unloading rate ˙PU. Hence, the

following expression is obtained by inserting the time derivative of Eq. (4.25) into Eq. (4.26): S =₋P˙H(t = τL+ τH)τU hmax + 2 h1/d max Λ B 1/d τU · lim t→ (τL+τH)+ ( M (t_{− (τ}L+ τH)) τU τT − (τL+ τH) τU 1/d) (4.27)

where M(0+₎ _{is the elastic indentation modulus M}

0. Then, Eq (4.27) is recast in S =₋P˙H(t = τL+ τH) τU hmax + 2 h1/d_max Λ B 1/d M0 (4.28)

where the second term of the right-hand side is the elastic contact stiffness S0 given by

Eq. (4.18). The discrepancy between S and S0 increases with the inverse of the

unload-ing penetration rate and, with the relaxation rate ˙PH at the end of the holding phase. As

shown by Fig. 4.3a, for a given deformable medium loaded at a given penetration rate, the longer the holding phase, the smaller the relaxation rate reached before release of the sustained penetration. Consequently, as shown in Fig. 4.3b, the longer the holding phase and the faster

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(a) History of force relaxation (b) Force-penetration curve

Figure 4.3: Effect of viscosity on the measurement of contact stiffness.

the unloading, the smaller the contact stiffness. If the relaxation reaches an asymptotic state ( ˙PH → 0) or, if the release of the indenter is infinitely fast (τU → 0), the viscous effects on

the measurement of the contact stiffness becomes negligible and Eq. (4.28) is equivalent to Eq. (4.18). Finally, the elastic indentation modulus is obtained by

M0= √ π 2pA_c S + ˙ PH(t = τL+ τH) τU hmax ! (4.29) where the contact stiffness S and the relaxation rate ˙PH at the end of the holding phase are

measured during the indentation experiment.

4.1.6 Correction of the analytical solution

As mentioned previously, the application of the analytical solution of Galin-Sneddon to solve a relaxation indentation problem with plastic deformations occurring during penetration of the indenter is an approximation. As a result, one should expect some bias in the indentation analysis leading to some inaccuracies of the assessed mechanical properties of the indented material. According to Oliver and Pharr (2004), the most important causes of this bias are:

• The non-consideration of large deformations; • The negligence of plastic deformations;

• The idealization of pyramidal indenters as axially symmetric probes.

It has been emphasized by Hay et al. (1999) that the assumption of small deformations is responsible for a mis-prediction of the radial component of the displacement field in compress-ible media. Meanwhile, the negligence of plastic deformations leads up to an overestimation of

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the elastic indentation modulus when evaluated with the BASh formula. Also, using the ana-lytical axisymmetric solution of Galin-Sneddon is an another approximation of the evaluation of the stress field for indentation with a pyramidal indenter (King, 1987).

A widely used method that consists in applying a correction factor β in Eq. (4.18) has been developed to counteract the biasing effects mentioned above. By doing so, Eq. (4.18) becomes

S0= β 2 √ π p AcM0. (4.30)

King (1987) was the first to emphasize the importance of this factor initially intended to correct for the idealization of non-circular probes. Later on, this method was used to correct for other biasing effects until it finally takes into account all types of physical processes that may bias the analytical solution of Galin-Sneddon (Oliver and Pharr, 2004). As a consequence of Eq. (4.30), the elastic indentation modulus given by Eq. (4.29) is recast in

M0 = √ π 2 βpA_c S + ˙ PH(t = τL+ τH) τU hmax ! (4.31) where β is taken equal to 1.034 for indentation with a Berkovich indenter and 1 when using a spherical probe King (1987).

4.1.7 Imperfect geometry of the indenter

In practice, the stiffness of the indenter is finite so that indentation after indentation, it deforms and deviates from the geometry described in Eq. (4.3). The consideration of these deviations is highly relevant as it strongly affects the accuracy of the results obtained by nanoindentation. In order to counteract this effect, Oliver and Pharr (2004) proposed a reformulation of the projected contact area used in Eqs. (4.29) and (4.31):

Ac= 8 X i=0 Ci {(hc)max}2 (1−i) = C0{(hc)max}2+· · · + C8{(hc)max}1/128 (4.32)

where Ci’s are constants to be calibrated from results of indentation performed on materials

with known mechanical properties. In case of undeformed conical indenters, Ac reduces to

C0{(hc)max}2 with C0 = π tan2θ. For spherical probes of radius R, the contact area is

C0{(hc)max}2+ C1(hc)max with C0 =−π and C1 = 2πR.

The maximum contact depth is expressed in terms of the maximum sinking depth illustrated in Fig. 4.4:

(hc)max= hmax− (hs)max. (4.33)

Oliver and Pharr (1992) express the maximum sinking depth in terms of the maximum applied load and the elastic contact stiffness:

(hs)max =

Pmax

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Figure 4.4: Schematic representation of an indentation (adapted from Oliver and Pharr (1992)).

where Pmax is measured at the onset of the holding phase (see Fig. 4.3a), and S0 is free of

viscous effects. The parameter is a constant determined from the Galin-Sneddon solution: = d + 1

d (1− Λ) = (

2 (1− 2/π) for d = 1

1 for d = 2 (4.35)

where Λ is given by Eq. (4.19).

From Eqs. (4.33) and (4.34), the projected contact area at maximum sustained penetration is linked to the maximum applied depth and measured load through the following equation:

Ac= 8 X i=0 Ci hmax− Pmax S0 2(1−i) (4.36) where assigning a value of 0.75 to has proven to be more successful than 2 (1 − 2/π) for Berkovich indenters (Oliver and Pharr, 1992). The elastic contact stiffness of Eq. (4.36) reads

S0 = S +

˙

PH(t = τL+ τH) τU

hmax (4.37)

where S is measured at the onset of unloading, ˙PH(t = τL+τH)is the relaxation rate measured

at the end of the holding phase and, hmaxand τU are prescribed parameters of the indentation

experiment.

4.2 Packing density distribution and strength properties

The hardness H is the average vertical traction component at the surface measured at the onset of the holding phase so that H ≡ P0/Ac where P0 stands for P (t = τL),

interchange-ably referred to as Pmax. Both the indentation modulus and hardness of C–S–H phases are

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phase and saturated pore space (Ulm et al., 2007; Vandamme et al., 2010). Then, assuming a nanogranular morphology along with the application of a self-consistent homogenization scheme with a solid percolation threshold at 50% porosity, the indentation modulus relates to the packing density as follows:

M0(ms, η) = ms(2η− 1) (4.38)

where ms is the indentation modulus of the solid phase and η is the packing density of the

indented microvolume. Although Eq. (4.38) is valid for a solid phase Poisson’s ratio of 0.2, it has a limited sensitivity to discrepancy from this value (Constantinides and Ulm, 2007). Meanwhile, a cohesive-frictional behavior of the solid phase is assumed so that the indentation hardness relates to the packing density η, the cohesion csand the friction coefficient αs of the

strength domain of hydrates (Vandamme et al., 2010) as follows: H(cs, αs, η) = csA h 1 + Bαs+ (Cαs)3+ (Dαs)10 i Π1+ αs(1− η)Π2 (4.39) where A = 4.7644, B = 2.5934, A = 2.1860 and D = 1.6777. In Eq. (4.39), Π1 is a function

of the packing density given by: Π1(η) = p 2(2η_{− 1) − 2η + 1} √ 2_{− 1} 1 + a(1_{− η) + b(1 − η)}2_{+ c(1}_{− η)}3 _(4.40)

where a = −5.3678, b = 12.1933 and c = −10.3071. Similarly, Π2 is a function of the friction

coefficient and packing density: Π2(αs, η) = 2η_{− 1} 2 d + e(1_{− η) + f(1 − η)α}s+ gα3s (4.41) where d = 6.7374, e = −39.5893, f = 34.3216 and g = −21.2053.

Therefore, an inverse analysis of the indentation results enables to assess the cohesion and friction coefficient of the solid phase of C–S–H, and the packing density at every indentation location within the specimen:

{cs, αs, η1, . . . , ηN} = argmin cs,αs,η1,...,ηN N X i=1 " M0,i− M(ηi) M0,i 2 + Hi− H(cs, αs, ηi) Hi 2# . (4.42) A number of indents N 2 is considered in Eq. (4.42), where the solid phase of C–S–H has a fixed indentation modulus ms of 63.5 GPa (Vandamme et al., 2010; Vandamme and Ulm,

2013).

4.3 Energy activated relaxation

The measured load relaxation of cement-based materials is assumed to result from the en-ergy activated Klug and Wittmann (1974); Wittmann (1982) rearrangement of nano-meter

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sized grains of C–S–H (Ulm et al., 2000; Jennings, 2004; Vandamme and Ulm, 2009, 2013). Therefore, in conformity with previous works of Klug and Wittmann (1974); Wittmann (1982); Jennings (2004), rate process theory is invoked to model the time-delayed mechanical response to indentation.

4.3.1 Simplification to a 1D problem

At any time of an indentation relaxation experiment, the stress field of the indented material is given by

σ(x, t) = Z t

0

C(t− τ) : [ ˙ε(x, τ) − ˙εp(x, τ )] dτ ∀ x ∈ Ω(t), ∀ t (4.43)

where ˙εp(t) is the time derivative of the plastic strain field at time t. Here, we consider that

all plastic deformations occur during the penetration of the indenter. We also assume that the loading phase is short enough so that no substantial relaxation occurs before the holding phase. If so, we have:

σ(x, t)≈ C(t − τL) : [ε(x, τL)− εp(x, τL)] ∀ x ∈ Ω(t), ∀ t s.t. τL≤ t ≤ τL+ τH. (4.44)

At the beginning of the holding phase, the material has not yet relaxed and the force applied is the largest load observed through the course of the experiment (see Fig. 4.3a):

Pmax ≡ P (τL) =

Z

∂Ωc(τL)

e_z_{· σ(x, τ}L)· n(x, τL) dA. (4.45)

Once the maximum penetration is sustained, the indented material starts relaxing and the applied force becomes

PH(t) = Z ∂Ωc(τL) e_z·{C(t − τL) : [ε(x, τL)− εp(x, τL)]}·n(x, τL) dA ∀ t s.t. τL≤ t ≤ τL+τH. (4.46) that we recast in PH(t) = Z ∂Ωc(τL) e_z_{· {C(t − τ}L) : S(0) : σ(x, τL)} · n(x, τL) dA ∀ t s.t. τL≤ t ≤ τL+ τH (4.47) where S(0) is the instantaneous elastic compliance of the solid. The indented material being isotropic, we have

S(0) = 1− 2ν0 E0 J +

1 + ν0

E0 (I− J) (4.48)

in which J = (1/3) 1⊗1 and 1 is the second-order unit tensor. Not only the indented material is elastic isotropic, but it remains isotropic during relaxation:

C(t) = E(t) 1_{− 2ν(t)} J +

E(t)

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An even stronger assumption is made that the Poisson’s ratio remains constant during relax-ation: C(t) = E(t) 1_{− 2ν}0 J + E(t) 1 + ν0 (I− J) . (4.50)

The statement of constant Poisson’s ratio is commonly made for the investigation of creep mechanisms in cement-based materials (Vandamme, 2008; Vandamme and Ulm, 2009; Jones and Grasley, 2011). The underlying idea at the origin of this assumption is that the relaxation of the solid is triggered by the relaxation of uniformly distributed microprestresses after a kinetic independent of external loadings (Bažant, 1997). As a result, every increment of stress released is coaxial with the applied stress state so that the Poisson’s ratio remains constant during relaxation.

As a result of these assumptions, we have PH(t) =

E(t− τL)

E0

Pmax ∀ t s.t. τL≤ t ≤ τL+ τH (4.51)

which, for larger times and short loading phases (t τL) becomes

PH(t) =

E(t) E0

Pmax ∀ t s.t. τL≤ t ≤ τL+ τH (4.52)

so that the kinetic of indentation load relaxation can be simplified to a 1D viscous problem. Given the following expression between the elastic indention and Young modulus:

M0= E0 1− ν2 0 (4.53) we have M (t) = φ(t)M0 (4.54)

where φ(t) ≡ PH(t)/Pmax is defined as a non-dimensional indentation relaxation function.

4.3.2 Rate process theory

Rate process theory was introduced by Eyring (1935) through a general equation governing the rate of rearrangement of matter for processes that can be idealized as the surmounting of a potential barrier by activated structural units. While the scope of the initial work of Eyring was to describe chemical reactions at molecular scale, it was extended to the representation of viscous flows (Eyring, 1936) and applied to the analysis of particulate flows as encountered in creep and relaxation of soil (Culling, 1983, 1988) and concrete (Klug and Wittmann, 1969, 1974). The corresponding rate equations used for activated rheological processes and the parameters related to nanostructural features involved in the mechanisms investigated in this study are presented here.

The general equation for the rate of a process in which matter rearranges by surmounting a potential energy barrier was proposed by Eyring (1935) in the form

κ = θ Fa Fn p m∗ (4.55)

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Displacement E n e rg y

Figure 4.5: Energy activated process of a flow unit.

where κ is the frequency of a primary process performed by an activated unit. The transmission coefficient θ is the probability that a unit having reached the activated state, referred to as transition state (Petersson, 2000), proceeds across the saddle point and completes the process rather than returning to its original configuration. The mean velocity p/m∗ _{of a unit along}

the axis of motion is given by the ratio of the mean momentum p over the reduced mass m∗_.

The ratio Fa/Fn is obtained from the partition function of the activated complex per unit

length Fa and the partition function of the normal state Fn. The different features involved

in such rearrangement processes are represented in Fig. 4.5, where Q is the energy barrier to surmount.

Eyring invokes statistical mechanics to reformulate Eq. (4.55) for the case of thermally acti-vated rearrangement processes with one translational degree of freedom:

κ = θ F_a∗ Fn kBT h exp − Q kBT (4.56) where κ is the rate of translational motion induced by thermal fluctuation. The energy avail-able being distributed among the system with respect to a Maxwell-Boltzmann distribution, the rate of activation is given by the proportion of units that bear enough energy to reach the transition state times the fundamental frequency kBT in which, kB is the Boltzmann’s

constant and h, the Planck’s constant. The partition function F∗

a differs from Fa in that it

is calculated using a zero of energy higher by Q than for Fn and, the partition function for

the degree of freedom normal to the barrier is omitted from F∗

a. By fixing Fa∗/Fn and θ to

unity (Eyring, 1936), Eq. (4.56) simplifies to κ = kBT h exp −_kQ BT . (4.57)

Following the notions of rate process theory applied to viscous flows (Eyring, 1936), viscosity is considered as the sliding of ordered layers of flow units at a distance λ1apart. The assumed

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Figure 4.6: Vacancy migration of flow units as described by the Eyring model applied to viscous flows.

mechanism consists in individual flow units which acquire the activation energy required to slip over the potential barrier to the next equilibrium position along the same plane (see Fig. 4.6). The viscosity of such a system is given by

η = σλ1/∆u (4.58)

where ∆u is the difference in velocity of two layers at a distance λ1 and σ is the applied

shearing stress. The distance from an equilibrium site to another is noted λ and the resulting force in the direction of motion is given by σλ2λ3, where λ2λ3 is the effective cross-sectional

area of the moving flow unit. As described in Fig. 4.7, the resulting force leads to a drop of σ V = σ/2(λλ2λ3) in the potential barrier of activation in direction of the applied stress so

that the rate of forward displacement is given by κ+ = kBT h exp − Q kBT − σ λλ2λ3 2 kBT . (4.59)

Similarly, the rate of backward translation is κ− = kBT h exp − Q kBT + σ λλ2λ3 2 kBT (4.60) so that the net frequency of forward displacement is

κ+ − κ− = 2 kBT h exp −_kQ BT sinh V σ kBT (4.61) where V , the thermodynamic activation volume, is the product of the distance crossed by an activated flow unit from equilibrium to transition state with the cross-sectional area of the sliding unit. Here, the transition state is located at equal distance from adjacent equilibrium sites so that the activation volume is given by 1/2λ · λ2λ3.

Although Eq. (4.61) does take into account the dispersion of energy in the system, it does not consider how free space is spread in the system. For a solid medium, not only a particle needs to be provided enough energy to slip over the potential barrier but, a sufficiently large volume needs to be free at proximity of a sliding unit in order for it to be able to move at a new equilibrium position. Therefore, the net frequency of activation is multiplied by a structural factor X that quantifies the effect of the distribution of room available on the viscosity of